Sensitivity analysis and optimization of eigenmode localization in continuum systems

Abstract

A model problem arising from optical design of photonic bandgap structure is investigated. That is, the optimization problem is to find the material inhomogeneity in a domain so that a particular eigenmode governed by the scalar Helmholtz equation is localized. The continuum sensitivity analysis of the objective function including the eigenmode is carried out. The derivative of the objective function with respect to the density function is obtained by the sensitivity problem and the adjoint problem in continuum systems. When the multiplicity of eigenmode happens, our strategy is to select the closest eigenmode to the current eigenmode. Four numerical examples in a square domain are studied, with different weight functions and initial density distributions. The numerical results illustrate the validity of the algorithm based on the continuum sensitivity analysis.